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a mathematical term that denotes the rule for determining a given distance between any two points (elements) of a given set A. A real number function that satisfies the following three conditions is called the distance ρ (a, b) between the points a and b of the set A: (1) ρ (a, b) ≧ 0, where ρ(a, b) = 0 if and only if a = b; (2) ρ(a, b) = ρ(b, a); and (3) ρ(a, b) + ρ(b, c) ≧ ρ(a, c). For a given set M, a metric may be introduced in other ways. For example, on a plane we may take not only the ordinary Euclidean distance
as the distance between points a and b having coordinates (x1, y1) and (x2, y2), respectively, but we may take other distances as well, for example,
ρ1(a, b) = ǀx1 – x2ǀ + ǀy1 – y2ǀ
In (functional and coordinate) vector spaces a metric is often defined as a norm or, sometimes, as a scalar product. In differential geometry, a metric is introduced by specifying an element of arc length by means of a differential quadratic form. A set that has a metric introduced in it is referred to as a metric space.
A metric is sometimes understood to denote a rule for determining not only distances but also angles; an example is a projective metric.
V. I. SOBOLEV